A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations

Transcription

1 A Hybridizable Discontinuous Galerkin Metod for te Compressible Euler and Navier-Stokes Equations J. Peraire and N. C. Nguyen Massacusetts Institute of Tecnology, Cambridge, MA 02139, USA B. Cockburn University of Minnesota, Minneapolis, MN 55455, USA In tis paper, we present a Hybridizable Discontinuous Galerkin (HDG) metod for te solution of te compressible Euler and Navier-Stokes equations. Te metod is devised by using te discontinuous Galerkin approximation wit a special coice of te numerical fluxes and weakly imposing te continuity of te normal component of te numerical fluxes across te element interfaces. Tis allows te approximate conserved variables defining te discontinuous Galerkin solution to be locally condensed, tereby resulting in a reduced system wic involves only te degrees of freedom of te approximate traces of te solution. Te HDG metod inerits te geometric flexibility and arbitrary ig order accuracy of Discontinuous Galerkin metods, but offers a significant reduction in te computational cost as well as improved accuracy and convergence properties. In particular, we sow tat HDG produces optimal converges rates for bot te conserved quantities as well as te viscous stresses and te eat fluxes. We present some numerical results to demonstrate te accuracy and convergence properties of te metod. I. Introduction Te numerical simulation of compressible flows as become an indispensable tool for many important applications suc as aero-acoustics, veicle design and turbomacinery. Altoug te ever increasing computer power allows us to solve complex problems tat would ave been intractable a few years ago, tere are still many problems of practical interest for wic te existing metods are inadequate. Terefore, te development of robust, accurate, and efficient metods for te numerical solution of te compressible Navier-Stokes equations in complex geometries remains a topic of considerable importance. In recent years, discontinuous Galerkin (DG) finite element metods ave emerged as a competitive alternative for solving nonlinear yperbolic systems of conservation laws. Te advantages of te DG metods over classical finite difference and finite volume metods are well-documented in te literature: 5, 6 te DG metods work well on arbitrary meses, result in stable ig order discretizations of te convective and diffusive operators, allow for a simple and unambiguous imposition of boundary conditions and are very flexible to parallelization and adaptivity. Despite all tese advantages, DG metods ave not yet made a significant impact for practical applications. Tis is largely due to te ig computational cost associated to DG metods wen compared to finite volume scemes. In tis paper, building on te previous work 7, 8, we develop a new class of ybridizable discontinuous Galerkin (HDG) metods for te numerical solution of te compressible Euler and Navier-Stokes equations. In addition to possessing local conservativity, ig-order accuracy, and strong stability for convectiondominated flows, te proposed HDG metods ave te following main advantages over many existing DG metods. First, unlike many oter DG metods (analyzed in Ref. [2]) wic result in a final system involving te degrees of freedom of te approximate field variables, te HDG metods produces a final system in terms Professor, Department of Aeronautics and Astronautics, M.I.T., 77 Massacusetts Avenue, AIAA Associate Fellow. Researc Scientist, Department of Aeronautics and Astronautics, M.I.T., 77 Massacusetts Avenue, AIAA Member. Professor, Scool of Matematics, University of Minnesota, Minneapolis. 1 of 11

2 of te degrees of freedom of te approximate traces of te field variables. Since te approximate traces are defined on te element faces only and single-valued on every face, te HDG metods ave significantly less te globally coupled unknowns tan oter DG metods. Tis large reduction in te degrees of freedom can lead to significant savings for bot computational time and memory storage. Second, te HDG metod exibits optimal convergence properties for te primal variables as well as teir fluxes. In fact, to our knowledge, it is te only DG metod tat acieves optimal convergence of te viscous fluxes in multidimensions. For te compressible Navier-Stokes equations, we sow results tat indicate tat te conserved quantities as well as te viscous stresses and eat fluxes, converge wit te optimal convergence of k + 1 in te L 2 - norm, wen polynomial approximations of order k are used. Finally, te HDG metods can deal wit inflow, outflow, slip, and solid wall boundary conditions weakly in a unified framework by defining appropriate numerical fluxes on te domain boundaries. Te HDG metods are devised by using te discontinuous Galerkin metodology to discretize te compressible Navier-Stokes equations wit appropriate coices of te numerical fluxes and by applying te ybridization tecnique to te resulting discretization. We first introduce te approximate traces of te conserved variables as new unknowns defined on te element boundaries. We ten define a total numerical flux (including bot te viscous and inviscid terms) in terms of te approximate traces and weakly impose te continuity of te normal component of te numerical flux. Tis results in a large nonlinear system of equations for te approximate field variables (including velocity, density, and energy and teir spatial derivatives) an te trace of te conserved variables. Tis nonlinear system is solved by using te Newton metod. After linearization, we can locally condense all te approximate field variables and teir spatial derivatives in an element-by-element fasion and obtain a reduced global system involving only te approximate traces of te conserved variables. Since te first DG metod 21 was introduced in 1973 by Reed and Hill for te transport equation, significant developments ave been made in te area of discontinuous Galerkin metods 10 for te numerical solution of nonlinear yperbolic conservation laws. Te first DG metod 3 for te compressible Navier- Stokes equations were proposed in 1997 by Bassi and Rebay. Sortly after, te local DG metod 9 and te Baumann-Oden DG metod 4 were developed for convection-diffusion problems. Based upon te work of Basi-Rebay 3 and Cockburn and Su, 9 Lomtev and Karniadakis developed a DG metod for te Navier- Stokes equations. 11 Te compact discontinuous Galerkin metod introduced in Ref. [18] ave also been applied to compressible viscous flows 19 and turbulent flows. 17 Recently, an interior penalty DG metod 20 was proposed by Hartmann and Houston for numerically solving te compressible Navier-Stokes equations. For tese DG metods, te use of explicit time-stepping metods proves very attractive since te resulting mass matrix is block-diagonal. However, wen an implicit time-stepping metod is used, tese DG metods produce a discrete system involving a large number of coupled degrees of freedom due to te nodal duplications along te interelement boundaries. Fortunately, tis major drawback of many DG metods is not present in te HDG metod since its globally coupled unknowns are defined on te element boundaries and are single-valued tere. Te paper is organized as follows. We introduce te HDG metod for te compressible Euler equations in Section 2 and in te compressible Navier-Stokes equations in Section 3. In eac section, we formulate te metod, briefly describe its implementation, and discuss te coice of te stabilization matrix and te treatment of te boundary conditions. In Section 4 we provide numerical results to assess te performance of te metod. Finally, in Section 5 we present some concluding remarks. II. HDG Metod for te Euler Equations A. Governing Equations and Notation We consider te steady-state Euler equations of gas dynamics defined over a domain Ω R d written in nondimensional conservation form as F (u) = 0, in Ω, (1) were u is te m-dimensional vector of conserved dimensionless quantities (namely, density, momentum and energy) and F (u) are te inviscid fluxes of dimension m d. Te Euler equations (1) must be supplemented wit appropriate boundary conditions at te inflow and outflow boundaries and at te solid wall. We discuss tese boundary conditions below. 2 of 11

3 To describe te HDG metod for solving te Euler equations, we introduce some notation. We denote by T a collection of disjoint regular elements K tat partition Ω and set T := { K : K T }. For an element K of te collection T, F = K Ω is te boundary face if te d 1 measure of F is nonzero. For two elements K + and K of te collection T, F = K + K is te interior face between K + and K if te d 1 measure of F is nonzero. We denote by E o and E te set of interior and boundary faces, respectively. We set E = E o E. Let P k (D) denote te space of polynomials of degree at most k on a domain D and let L 2 (D) be te space of square integrable functions on D. We introduce te following discontinuous finite element approximation space W k = {w (L 2 (T )) m : w K (P k (K)) m, K T }. In addition, we introduce a finite element approximation space for te approximate trace of te solution M k = {µ (L 2 (E ) m : µ F (P k (F )) m, F E }. Note tat M consists of functions wic are continuous inside te faces (or edges) F E and discontinuous at teir borders. Finally, we define various inner products for our finite element spaces. We write (w, v) T := K T (w, v) K, were (w, v) D denotes te integral of wv over te domain D R d for w, v P. We also write (w, v) T := m i=1 (w i,v i ) T for w, v W k. We ten write η, ζ T := K T η, ζ K and η, ζ T := m i=1 η i,ζ i T, for η, ζ M k, were η, ζ D denotes te integral of ηζ over te domain D R d 1. B. Formulation of te HDG Metod We seek an approximation u W k suc tat for all K T, (F (u ), w) K + F n, w = 0, w (P k (K)) m. (2) Here, te numerical flux F is an approximation to F (u) over K. We take te numerical flux of te form K F n = F (û ) n + S(u, û )(u û ), on K, (3) were û M k is an approximation to te trace of te solution u on K, and S(u, û ) is a local stabilization matrix wic as an important effect on bot te stability and accuracy of te resulting sceme. Alternative definitions for te numerical flux are given in Ref. [13]. Te selection of te matrix S will be described below. By adding te contributions of (2) over all te elements and enforcing te continuity of te normal component of te numerical flux, we arrive at te following problem: find an approximation (u, û ) W k M k suc tat (F (u ), w) T + F n, w = 0, w W k, T F n, µ + B, µ = 0, µ M k. (4) T \ Ω Here B is te numerical flux vector of dimension m and is defined over te boundary Ω. Its precise definition depends on te types of boundary conditions and will be given below. Note tat û is singlevalued over eac edge since û belongs to M k. Furtermore, we note tat even toug te quantity [ F n] may not be zero pointwise, its projection is, tat is, P[ F n] = 0, were P denotes te L 2 projection into M k. Terefore, our sceme is conservative since only te projection of F n enters te first equation in (4). By applying te Newton-Rapson procedure to solve te weak formulation (4) we obtain at every Newton step a matrix system of te form [ ]( ) ( ) A B U F =, C D Λ G were U and Λ are te vectors of degrees of freedom of u and û, respectively. It is important to note tat te matrix A as block-diagonal structure. Terefore, we can eliminate U to obtain a reduced system in terms of Λ as K Λ= R, Ω 3 of 11

4 were K = D CA 1 B and R = G CA 1 F. Tis is te global system to be solved at every Newton iteration. Since û is defined and single-valued along te faces, te final matrix system of te HDG metod is smaller tan tat of many oter DG metods. Moreover, te matrix K is compact in te sense tat only te degrees of freedom between neigboring te faces tat sare te same element are connected. C. Stabilization Matrix We propose ere two scemes to define te stabilization matrix. In te first sceme, we coose S = L Λ R, (5) were L, R, and Λ are te matrices of te left and rigt eigenvectors, and eigenvalues of te Jacobian matrix [ F (û )/ û ] n, respectively. Te second sceme inspired by te local Lax-Friedric metod involves coosing S = λ max I, (6) were λ max is te maximum absolute value of te eigenvalues of te matrix Λ, and I is te identity matrix. Te proposed scemes appear to be new as te stabilization matrix depends only on one state defined over te element interface. In contrast, traditional finite volume and DG metods define te stabilization matrix as a function of two states defined over two opposite sides of te element interface. D. Numerical Fluxes In order to gain some insigt into te form of te numerical fluxes and to be able to compare te numerical fluxes used ere wit te more standard forms used by oter DG metods, we insert te expression (3) into te second equation in (4). If we assume tat te stabilization matrix is constant over a face ten we obtain te following expressions for û and F n, û = 1 2 (ul + u R ), F n L = F (û ) n L S(uR u L ). (7) Here, u L and ur denote te left and rigt DG approximations at eiter side of te interface and nl is te unit normal to te interface pointing from left side to te rigt side. We observe tat wen te stabilization matrix is cosen according to expression (6), ten tis sceme reduced precisely to te local Lax-Friedrics sceme. Wen te stabilization matrix is cosen according to (5), ten te resulting sceme resembles Roe s sceme but wit some differences. E. Boundary Conditions 1. Inflow/Outflow boundary conditions At te inlet section or outlet section of te flow, we need to set te state variable u to te freestream condition u. To tis end, we define te boundary flux vector B as B = A + n (û )(u û ) A n (û )(u û ), (8) were A n = A n and A ± n =(A n ± A n )/2. Here A n =[ F / u] n denotes te Jacobian of te inviscid normal flux to te boundary. 2. Slip boundary condition At te solid surface wit slip condition, we must impose zero normal velocity. Hencefort, we set were te vector b is defined in terms of u as follows B =(b û ), (9) b [1] = u [1], b [2,..., m 1] = v nv n, b [m] =u [m]. (10) Here v = u [2,..., m 1] are te velocity components of u and v n = v n is te normal component of te approximate velocity. Note tat since (v nv n ) n = 0 we ave v n = 0, were v = û [2,..., m 1] are te velocity components of û. 4 of 11

5 F. Extension to te Unsteady Euler Equations We end tis section by extending te HDG metod to te unsteady Euler equations u t + F (u) = 0, in Ω (0,t f ], u = u 0, on Ω {t =0}. Te boundary conditions are te same as te steady-sate case. Using te Backward-Euler sceme at time level t j wit timestep t j we find an approximation (u j, ûj ) W k M k suc tat ( u j ) ) t, w (F (u j ), w T + F j n, µ T \ Ω + F j T n, w = Bj Ω, µ ( u j 1 ) t, w, w W k, (11) = 0, µ M k, were F j n = F (ûj ) n + S(uj, ûj )(uj ûj ), (12) and te boundary numerical flux B j is already described earlier. Tis discrete system is similar to te system (4) for te steady-state case except tat tere are two additional terms due to te backward difference discretization of te time derivative. We can tus apply te same solution procedure described above for te steady-state case to te time-dependent case at every time level. Since using iger-order BDF scemes or diagonally implicit Runge-Kutta (DIRK) metods would yield a discrete system similar to (11), te HDG metod for spatial discretization can also be used wit tese implicit ig-order time-stepping scemes. Tis leads to a ig-order accurate metod in bot space and time. III. HDG Metod for te Navier-Stokes Equations A. Governing Equations We consider te steady-state compressible Navier-Stokes equations written in conservation form as q u = 0, in Ω, (F (u) + G(u, q)) = 0, in Ω, (13) were G(u, q) are te viscous fluxes of dimension m d. Te nondimensional form of te Navier-Stokes equations as well as te definition of te inviscid and viscous fluxes can be found in Ref. [1]. Of course, te Navier-Stokes equations (13) sould be supplemented wit appropriate boundary conditions at te inflow, outflow and solid wall boundaries, as well as a source term wic is omitted for simplicity of exposition. We discuss below ow to deal wit tese boundary conditions. In addition to te notation introduced in Section II, we need to define a new approximation space as V k = {v (L 2 (T )) m m : v K (P k (K)) m m, K T }. Te approximate gradient q, wic approximates q, resides in tis space. B. Formulation Following te metod of line for te Euler equations we seek an approximation (q, u, û ) V k W k M k suc tat (q, v) T +(u, v) T û, v n T = 0, v V k, (F (u )+G(u, q ), w) T + ( F + Ĝ) n, w = 0, w W k, T ( F (14) + Ĝ) n, µ + B, µ = 0, µ M k. T \ Ω Ω 5 of 11

6 Here, te numerical fluxes F and Ĝ are an approximation to F (u) and G(u, q) over K, respectively. In addition, B is te numerical flux vector of dimension m defined over te boundary. As before, we take te interior numerical fluxes of form ( F + Ĝ) n =(F (û )+G(û, q )) n + S(u, û )(u û ), (15) were te stabilization matrix S can be selected by using te same expressions proposed above for te Euler equations. Te boundary flux vector B depends on te types of boundary conditions, te precise definition of wic will be described below. By applying te Newton-Rapson procedure to solve te nonlinear system (14) we obtain at every Newton step a matrix system of te form A B E C D L M N P were Q, U and Λ are te vectors of degrees of freedom of q, u and û, respectively. We note tat te degrees of freedom for q, u are grouped togeter and ordered in an element-wise fasion, te corresponding matrix [A B; CD] as block-diagonal structure. Te size of eac block is given by te number of degrees of freedom of q, u associated to eac element. Terefore, we can eliminate bot Q and U to obtain a reduced system in terms of Λ as A Λ= F, were [ A = P M ] [ ] 1 [ A B N C D E L ] Q U Λ = [, F = G H F G M, ] [ ] 1 [ A B N C D Te matrix A as te same size and structure as te matrix K of te HDG metod for te Euler equations. C. Wall Boundary Conditions At te solid surface wit no slip, we impose zero velocity and eiter a fixed temperature T = T w (isotermal wall) or zero eat flux T/ n = 0 (adiabatic wall). First, we note tat te first component of bot F (u) n and G(u, q) n vanises at te solid wall since u is zero tere. Terefore, we must set te first component of B equal to te first component of F + Ĝ wic will be set to zero weakly. We ten set te next m 2 (velocity) components of B to be equal to te velocity components of û, wic in turn weakly enforces te velocity components of û to be zero at te wall. Te last component of B depends on weter te wall is isotermal or adiabatic. For te adiabatic wall, we set te last component of B equal to te last component of F + Ĝ since te last component of bot F (u) n and G(u, q) n vanises at te solid wall. For te isotermal wall, we set te last component of B to T w T, were T is te approximate trace of te temperature and determined from û. D. Extension to te Unsteady Navier-Stokes Equations Finally, we extend te HDG metod to te unsteady Navier-Stokes equations q u = 0, in Ω (0,t f ], u t + (F (u) + G(u, q)) = 0, in Ω (0,t f ], u = u 0, on Ω {t =0}. Te boundary conditions are te same as te steady-state case. H F ]. 6 of 11

7 Figure 1. Inviscid flow over a Kármán-Trefftz airfoil: M =0.1, α =0. Detail of te mes employed (left) and Mac number contours of te solution using fourt order polynomial approximations (rigt). For simplicity of exposition we use te Backward-Euler sceme to discretize te time derivative wit timestep t j for j 1. At time level t j we seek an approximation (q j, uj, ûj ) V k W k M k suc tat ) (q j, v +(u j, v) T û j, v n = 0, w V k, T T ( u j ) ) t, w (F (u j )+G(uj, qj ), w + ( F ( u j j 1 ) + Ĝj ) n, w = T T t, w, w W k (, F j + Ĝj ) n, µ + Bj, µ = 0, µ M T \ Ω Ω k. (16) As before, we define te interior numerical fluxes as ( F j + Ĝj ) n =(F (ûj )+G(ûj, qj )) n + S(uj, ûj )(uj ûj ), (17) and te boundary numerical flux B j as already described in te previous subsection. Since tis discrete nonlinear system is similar to te system (14) for te steady-state case, we apply te same solution procedure described above for te steady-state case to te time-dependent case at every time level. IV. Numerical Results We sow in tis section some representative computations carried out wit te algoritms described above. All te computations reported ave been done wit te stabilization matrix given by equation 5, but we ave found no appreciable differences wen using te alternative simpler stabilization form given by equation 6. A. Euler flow Te first example we consider is tat of an inviscid flow over an airfoil. Te geometry of te airfoil is obtained by transforming a circle in te complex ζ-plane into te complex z-plane using a Kármán-Trefftz trasnformation ( ) n z n ζ 1 z + n = (18) ζ +1 For te particular airfoil considered ere te circle on te ζ-plane is centered at point ( 0.05, 0.05) and te radius of te circle is suc tat it passes troug te point (1, 0). Te value of n determines te trailing edge angle β t, and for our example it is cosen as n =2 β t /π =1.98. Te airfoil geometry as well as a detail of te mes utilized in all our computations is sown in figure 1. Te figure also sows te Mac number contours obtained for a free stream Mac number M =0.1 and an angle of attac α = 0 using fourt order polynomial approximations (k = 4). Te distribution of pressure coefficient and entropy deviation s =(p/p )/(ρ/ρ ) γ over te airfoil surface is sown in figure 2 for polynomial approximations k =2, 3 and 4. Wile tere is no appreciable difference in te pressure coefficient, te entropy deviation is reduced considerably every time te order of approximation is increased. 7 of 11

8 !"(!"& k =2 k =3 k = k =2 k =3 k =4!"$ Cp!!!"$!!"&!!"( Entr op ydevi ati on !!"* 0.997!# 0.996!#"$!!"#!"$!"%!"&!"'!"(!")!"*!"+ # x/c x/c Figure 2. Inviscid flow over a Kármán-Trefftz airfoil: M =0.1, α =0. Pressure coefficient distribution (left) and entropy deviation (rigt) over te airfoil surface.!"(!"& k =4 Exact!"$!!!"$ Cp!!"&!!"(!!"*!#!#"$!!"#!"$!"%!"&!"'!"(!")!"*!"+ # x/c Figure 3. Inviscid flow over a Kármán-Trefftz airfoil at α =0. Comparison of te pressure coefficient between te HDG solution for M =0.1 using a k =4approximation and te exact incompressible potential solution. In order to validate tis solution, we sow in figure 3 te comparison of te pressure coefficient calculated wit k = 4 and te analytical potential solution for te incompressible case. Te very minor differences between te two solutions can be attributed to compressibility effects as well as to te proximity of te far field boundary in te Euler computations. In te computations, te far field boundary as only been placed at a distance of five cords away from te airfoil and no vortex correction as been applied. Note tat even in tis case, te errors are expected to be small since te angle of attack considered is zero and ence te lift is small wic in turn implies tat te first order vortex correction would also be small. B. Couette Flow Tis example is aimed at verifying te accuracy and convergence of te metod for te Navier-Stokes equations. We consider a compressible Couette flow wit a source term on a square domain (0 x H and 0 y H). Te exact solution is given by u = Uy log (1 + y), v =0, p = p and T = T 0 + y(t 1 T 0 )+ γ 1 2γp Pry (1 y), were y = y/h. Te density is is ten given by ρ =1/T. Te source term is determined from te exact solution as ( s = 0, y +2 ) (y + 1) 2, 0, 3 log(y + 1) + y(2 log(y + 1) + 2) + 1 (y + 1) 2 3 log(y + 1) log(y + 1) 2. 8 of 11

9 degree mes ρ ρ T p p T ρe ρe T τ τ T T k 1/ error order error order error order error order error order e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 1. History of convergence of te HDG metod for te compressible Couette flow wit a source term. For our numerical experiments we take U = 1, H = 1, T 0 =0.8and T 1 =0.85, Pr =0.72 and p =1/(γM ) 2 wit M =0.15. Te computational domain is discretized into a uniform square mes and ten eac square is furter divided into two triangles. In order to asses te accuracy of te computed solution we calculate te L 2 -norm of te error in te density ρ, te linear momentum vector p =(ρu, ρv), te energy ρe, te eat flux = T, and te stress tensor τ = ( 2 3 (2u x v y )+p u y + v x u y + v x 2 3 (2v y u x )+p for different mes sizes and different polynomial orders. Table 1 sows te computed errors and orders of convergence for te approximations to tese quantities. Note ere tat te approximate stresses τ and eat flux are computed based on te approximate conserved variables u and te approximate gradients q. We observe tat te L 2 -norm of te error in tese approximate quantities converges wit te optimal order of accuracy of k + 1. C. Navier-Stokes flow Te final example we consider is a laminar flow over an airfoil. Te geometry of te airfoil and te mes employed is te same tat as been used for te Euler example. Te caracteristics of te flow are M =0.1, Re =4, 000 and Pr =0.72. Figure 4 sows te Mac contours of te solution computed using a fourt order polynomial approximation. Also sown in te figure is a detail of te mes and solution near te leading edge. were one can observed tat a ig order element is actually sufficient to cleanly capture te boundary layer. In order to verify tat tis is a grid converged solution we sow in figure 5 te distribution of pressure and skin friction coefficient using polynomial approximations of k =2, 3 and 4. It can be readily observed tat differences in te surface quantities computed for te different meses are very small. ) 9 of 11

10 Figure 4. Inviscid flow over a Kármán-Trefftz airfoil: M =0.1, Re = 4000 and α =0. Mac number distribution (left) and detail of te mes and Mac number solution near te leading edge region (rigt) using fourt order polynomial approximations.!"&!"$!"%!"$ k =2 k =3 k =4!!!"$!"# Cp!!"&!!"( k =2 k =3 k =4 Cf!!!"#!!"*!#!!"$!#"$!!"%!#"&!!"#!"$!"%!"&!"'!"(!")!"*!"+ # x/c!!"&!!"#!"$!"%!"&!"'!"(!")!"*!"+ # x/c Figure 5. Laminar flow over a Kármán-Trefftz airfoil: M =0.1, Re = 4000 and α =0. Pressure coefficient distribution (left) and skin friction coefficient (rigt) over te airfoil surface. V. Conclusions We ave presented a ybridizable discontinuous Galerkin metod for te numerical solution of te compressible Euler and Navier-Stokes equations. Te proposed metod olds important advantages over many existing DG metods in terms of te globally coupled degrees of freedom and in te improved accuracy. Moreover, te HDG metod is somewat simpler to implement tan oter implicit DG metods and allows for te simple treatment of boundary conditions and numerical fluxes. Te numerical results sow tat te HDG metod is efficient for te numerical simulation of inviscid and viscous compressible flows. Our current researc is focused on te development of efficient iterative metods for solving te linear system wic arises from application of te Newton-Rapson procedure. Acknowledgments J. Peraire and N. C. Nguyen would like to acknowledge te Singapore-MIT Alliance and te Air Force Office of Scientific Researc under te MURI project on Biologically Inspired Fligt for partially supporting tis work. B. Cockburn would like to acknowledge te National Science Foundation for partially supporting tis work troug Grant DMS References 1 D. A. Anderson, J. C. Tanneill, and R. H. Pletcer. Computational Fluid Dynamics and Heat Transfer. Hemispere Publising, New York, D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin metods for elliptic 10 of 11

11 problems. SIAM J. Numer. Anal., 39(5): , F. Bassi and S. Rebay. A ig-order accurate discontinuous finite element metod for te numerical solution of te compressible Navier-Stokes equations. J. Comput. Pys., 131: , C. Baumann and J. Oden. A discontinuous p finite element metod for convection-diffusion problems. Comput. Metods Appl. Mec. Engrg., 175: , B. Cockburn. Discontinuous Galerkin metods. ZAMM Z. Angew. Mat. Mec., 83: , B. Cockburn. Discontinuous Galerkin Metods for Computational Fluid Dynamics. In R. de Borst E. Stein and T.J.R. Huges, editors, Encyclopedia of Computational Mecanics, volume 3, pages Jon Wiley & Sons, Ltd., England, B. Cockburn, J. Gopalakrisnan, and R. Lazarov. Unified ybridization of discontinuous Galerkin, mixed and continuous Galerkin metods for second order elliptic problems. SIAM J. Numer. Anal., 47: , B. Cockburn, J. Gopalakrisnan, N. C. Nguyen, J. Peraire, and F-J Sayas. Analysis of an HDG metod for Stokes flow. Mat. Comp.. To appear. 9 B. Cockburn and C. W. Su. Te local discontinuous Galerkin metod for convection-diffusion systems. SIAM J. Numer. Anal., 35: , B. Cockburn and C.-W. Su. Runge-Kutta discontinuous Galerkin metods for convection-dominated problems. J. Sci. Comput., 16: , I. Lomtev and G. E. Karniadakis. A discontinuous Galerkin metod for te Navier-Stokes equations. International Journal for Numerical Metods in Fluids, 29: , N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit ig-order ybridizable discontinuous Galerkin metod for linear convection-diffusion equations. J. Comput. Pys., 228: , N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit ig-order ybridizable discontinuous Galerkin metod for nonlinear convection-diffusion equations. J. Comput. Pys., 228: , N.C. Nguyen, J. Peraire, and B. Cockburn, A ybridizable discontinuous Galerkin metod for Stokes flow. Computer Metods in Applied Mecanics and Engineering. To appear. 15 N.C. Nguyen, J. Peraire, and B. Cockburn, A comparison of HDG metods for Stokes flow. Submitted. 16 N.C. Nguyen, J. Peraire, and B. Cockburn, A ybridizable discontinuous Galerkin metod for te incompressible Navier- Stokes equations. Submitted. 17 N.C. Nguyen, P.O. Persson, and J. Peraire, RANS Solutions using ig order discontinuous Galerkin metods. In Proceedings of te 45t AIAA Aerospace Sciences Meeting and Exibit, AIAA , Reno, NV, January J. Peraire and P. O. Persson. Te compact discontinuous Galerkin (CDG) metod for elliptic problems. SIAM Journal on Scientific Computing, 30(4): , P. O. Persson and J. Peraire. Newton-GMRES preconditioning for discontinuous Galerkin discretizations of te Navier- Stokes equations. SIAM Journal on Scientific Computing, 30(6): , R. Hartmann and P. Houston An optimal order interior penalty discontinuous Galerkin discretization of te compressible NavierStokes equations. J. Comput. Pys., 227: , W.H. Reed and T.R. Hill. Triangular mes metods for te neutron transport equation. Tecnical Report LA-UR , Los Alamos Scientific Laboratory, C. Liang, A. Jameson and Z.J. Wang. Spectral difference metod for compressible flow on unstructured grids wit mixed elements. J. Comp. Pys., 228: , of 11